Molecular closure approximations

VI. Beyond Thermodynamic Perturbation Theory Molecular Closure Approximations... 

VI. BEYOND THERMODYNAMIC PERTURBATION THEORY MOLECULAR CLOSURE APPROXIMATIONS... 

Taken as a whole, the ideas discussed led Yethiraj and Schweizer to propose the following reference molecular closure approximations for site interaction potentials consisting of a hard core plus tail" ... 

Structure and the molecular closure approximations. Very recent work by Gromov and de Pablo has shown for the symmetric blend model that PRISM with the R-MPY closure is in excellent agreement with continuous space simulations for the structure, mixing thermodynamic properties, and the coexistence curve. 

The development of molecular closure approximations was guided by three considerations (i) the indirect two-molecule correlation processes were explicitly included in the closure relation, (ii) contributions of the hard core and attractive tail parts of the potential were treated via separate approximations, and (iii) the closures recover exactly the two-molecule correlation processes in the high temperature weak coupling limit. The simplest approximation which incorporates the above three points is... 

This same idea is used in the development of molecular closure approximations by Sch weizer and Yethiraj [90]. The hard core reference system is treated with the PY closure. The closure for the attractive tail part of the potential is formulated to exactly describe the weak coupling limit. The simplest molecular closure that correctly treats the longer ranged potentials (i.e., gives Rory-Huggins scaling) is given by... 

The second high-frequency term involves a sum over all discrete states and an integration over the continuum states the difficulties involved have been outlined before. Little is known about the continuum states, but what few calculations there are for simple systems92 suggest that they may be at least as important as the discrete states. For this reason early calculations were done in the closure approximation, notably by Van Vleck in the 1930 s. The difficulties of calculating xHF have been reviewed by Weltner.93 Experimentally xHF may be obtained from rotational magnetic moments. For linear molecules these can be obtained from molecular-beam experiments, which also measure the anisotropy x Xi- directly. The anisotropies may also be derived from crystal data, the Cotton-Mouton effect and, recently, Zeeman microwave studies principally by Flygare et al.9i... 

For a system with electrostatic interactions, it is appropriate to complement Eq. (4.57) by the molecular HNC closure approximation, which relates the 3D correlations between the ion and the entire solvent molecule. This provides consistency of the solvent-ion correlations and, in particular, ensures the steric constraints for the ion-solvent site distributions. The solvent-ion SC-3D-RISM/HNC equations for the 3D solvent-ion TCFs h r) can be cast in the form... 

The simplest molecular closure based on the above ideas is one that builds in the hard core reference behavior and correctly treats the longer ranged attractive potentials in the weak coupling limit. It is called the Reference Molecular Mean Spherical Approximation (RMMSA) and is given in real space for a homopolymer blend by [68-70]... 

Extensive analytic results for the symmetric thread blend have also been derived [68,70b]. In the thread-polymer limit the hard core condition becomes irrelevant for the molecular closure relations. In particular, for the R-MMSA and R-MPY/HTA approximations the MM (k) functions are fully specified by the closure relations, and their k = 0 values are given in general by... 

Alternative closure approximations for the repulsive force fluid have been investigated and will be briefly commented on in subsequent sections. Based on the idea that the atomiclike closures are useful by analogy for molecular fluids, there are several alternatives to the PY or MSA for hard core fluids. These include the hypernetted chain (HNC) approximation... 

The strategy for explicitly formulating the molecular closures was guided by three considerations." (1) Use of the commonly employed reference approach. The successful site-site PY closure is retained to describe the repulsive force reference fluid but a molecular closure scheme is adopted to describe the attractive, slowly varying forces. (2) The approximation scheme is required to provide an exact description of the structural consequences of the tail potentials at the two-molecule level in the weak coupling limit [/3umm-W 1]- (3) Use of an appropriate site-site approximation for the direct attractive interaction contribution motivated by experience in simple fluids. ... 

The effect of attractions on the structure of dense one-component polymer melts. According to the van der Waals ideas, attractions should have very little effect. Surprisingly, we are unaware of simulations that have probed this question, although they are now in progress. Recent PRISM studies by Butler and Schweizer using atomic and molecular closures have been carried out. Repulsive force screening of the effects of attractions on structure is recovered for many, but not all, closure approximations. 

The reduction of thread PRISM with the R-MMSA closure for the idealized fully symmetric block copolymer problem to the well-known incompressible RPA approach " is reassuring. However, in contrast with the blend case, for copolymers that tend to microphase separate on a finite length scale, the existence of critical or spinodal instabilities is expected to be an artifact of the crude statistical mechanical approximations. That is, finite N fluctuation effects are expected to destroy all such spinodal divergences and result in only first-order phase transitions in block copolymers [i.e., Eq. (7.3) is never satisfied]. Indeed, when PRISM theory is numerically implemented for finite thickness chain models using the R-MMSA or R-MPY/HTA closures spinodal divergences do not occur. Thus, one learns that even within the simpler molecular closures, the finite hard-core excluded volume constraint results in a fluctuation effect that destroys the mean-field divergences. 

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